#913 ⟨a, b | aa=b, abbb=b⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6
11aa2a3a4a5a6
aaa2a3a4a5a6a2
a2a2a3a4a5a6a2a3
a3a3a4a5a6a2a3a4
a4a4a5a6a2a3a4a5
a5a5a6a2a3a4a5a6
a6a6a2a3a4a5a6a2

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a7 ⇒ a2 [2]
2. b ⇒ a2 [1]
# ab:aa=b,abbb=b a/b - -
aaaaaaa=aa
b=aa

Same cardinality

22 unique, 1141 total

Σ#PresentationPropertiesDescriptionφ
7116⟨a, b | aaa=b, abb=1⟩Grp Com 7ℤ7763
8577⟨a, b | aaa=b, abb=a⟩Noc Com 7ℕ(7 = 1)59
8578⟨a, b | aaa=b, abb=b⟩Noc Com 7ℕ(7 = 3)44
8937⟨a, b | ab=a, baaa=b⟩Noc 766
91593⟨a, b | aaa=ab, baa=b⟩Noc 73
91601⟨a, b | aaa=bb, aab=b⟩Noc Com 75
91620⟨a, b | aab=aa, bbb=a⟩Noc Com 7ℕ(7 = 6)35
91632⟨a, b | aab=ab, bbb=a⟩Noc Com 7ℕ(7 = 4)50
92074⟨a, b | aab=b, babb=a⟩Noc Com 720
92231⟨a, b | ab=aa, aaa=bb⟩Noc 73
92271⟨a, b | bb=aa, aaa=ab⟩Noc 73
93197⟨a, b | ab=a, bbb=baa⟩Noc 75
104162⟨a, b | abb=aab, bbb=a⟩Noc Com 7ℕ(7 = 5)42
106251⟨a, b | aaa=a, aabba=b⟩Noc 71
108987⟨a, b | ab=a, aaaaa=bb⟩Noc Com 75
109108⟨a, b | ab=a, bbbbb=aa⟩Noc Com 72
109110⟨a, b | ab=a, bbbbb=ba⟩Noc 71
109263⟨a, b | ab=a, aaaa=bbb⟩Noc Com 74
109375⟨a, b | ab=a, bbba=bbb⟩Noc 73
109376⟨a, b | ab=a, bbbb=aaa⟩Noc Com 73
109382⟨a, b | ab=a, bbbb=bba⟩Noc 71
1114843⟨a, b | abba=b, aabab=a⟩Noc 71

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

82 total

Σ#PresentationMapping
8917⟨a, b | aa=b, babb=b⟩φ(a) = a, φ(b) = aa
91600⟨a, b | aaa=bb, aab=a⟩φ(a) = aaaa, φ(b) = a
91656⟨a, b | aab=bb, bbb=a⟩φ(a) = aaa, φ(b) = a
91684⟨a, b | aba=bb, bbb=a⟩φ(a) = aaa, φ(b) = a
92903⟨a, b | aa=b, aaabb=b⟩φ(a) = a, φ(b) = aa
92907⟨a, b | aa=b, aabab=b⟩φ(a) = a, φ(b) = aa
92909⟨a, b | aa=b, aabba=b⟩φ(a) = a, φ(b) = aa
92913⟨a, b | aa=b, abaab=b⟩φ(a) = a, φ(b) = aa
92915⟨a, b | aa=b, ababa=b⟩φ(a) = a, φ(b) = aa
92925⟨a, b | aa=b, baaab=b⟩φ(a) = a, φ(b) = aa
93058⟨a, b | aa=b, abbb=aa⟩φ(a) = a, φ(b) = aa
93065⟨a, b | aa=b, babb=aa⟩φ(a) = a, φ(b) = aa
105196⟨a, b | aab=bb, aabb=a⟩φ(a) = aaa, φ(b) = a
105200⟨a, b | aab=bb, abab=a⟩φ(a) = aaa, φ(b) = a
105208⟨a, b | aab=bb, baab=a⟩φ(a) = aaa, φ(b) = a
105280⟨a, b | aba=bb, aabb=a⟩φ(a) = aaa, φ(b) = a
105282⟨a, b | aba=bb, abab=a⟩φ(a) = aaa, φ(b) = a
105284⟨a, b | aba=bb, abba=a⟩φ(a) = aaa, φ(b) = a
105288⟨a, b | aba=bb, baab=a⟩φ(a) = aaa, φ(b) = a
108643⟨a, b | aa=b, aaaaab=b⟩φ(a) = a, φ(b) = aa
108645⟨a, b | aa=b, aaaaba=b⟩φ(a) = a, φ(b) = aa
108649⟨a, b | aa=b, aaabaa=b⟩φ(a) = a, φ(b) = aa
108923⟨a, b | aa=b, aaabb=aa⟩φ(a) = a, φ(b) = aa
108930⟨a, b | aa=b, aabab=aa⟩φ(a) = a, φ(b) = aa
108934⟨a, b | aa=b, aabba=aa⟩φ(a) = a, φ(b) = aa
108942⟨a, b | aa=b, abaab=aa⟩φ(a) = a, φ(b) = aa
108946⟨a, b | aa=b, ababa=aa⟩φ(a) = a, φ(b) = aa
108964⟨a, b | aa=b, baaab=aa⟩φ(a) = a, φ(b) = aa
1112406⟨a, b | aaba=bb, abaa=a⟩φ(a) = aa, φ(b) = a
1112414⟨a, b | aaba=bb, baaa=a⟩φ(a) = aa, φ(b) = a
1112496⟨a, b | abaa=bb, baaa=a⟩φ(a) = aa, φ(b) = a
1112612⟨a, b | abbb=bb, bbbb=a⟩φ(a) = aaaa, φ(b) = a
1112638⟨a, b | babb=bb, bbbb=a⟩φ(a) = aaaa, φ(b) = a
1115546⟨a, b | aab=aa, aaaaa=b⟩φ(a) = a, φ(b) = aaaaa
1115739⟨a, b | aab=bb, aaaab=a⟩φ(a) = aaa, φ(b) = a
1115802⟨a, b | aba=aa, aaaaa=b⟩φ(a) = a, φ(b) = aaaaa
1115907⟨a, b | aba=bb, aaaab=a⟩φ(a) = aaa, φ(b) = a
1115909⟨a, b | aba=bb, aaaba=a⟩φ(a) = aaa, φ(b) = a
1115913⟨a, b | aba=bb, aabaa=a⟩φ(a) = aaa, φ(b) = a
1119307⟨a, b | aaa=b, aaaab=aa⟩φ(a) = a, φ(b) = aaa
1119311⟨a, b | aaa=b, aaaba=aa⟩φ(a) = a, φ(b) = aaa
1119319⟨a, b | aaa=b, aabaa=aa⟩φ(a) = a, φ(b) = aaa
1119383⟨a, b | aab=a, aaaab=bb⟩φ(a) = aaaa, φ(b) = a
1119387⟨a, b | aab=a, aaaba=bb⟩φ(a) = aaaa, φ(b) = a
1119395⟨a, b | aab=a, aabaa=bb⟩φ(a) = aaaa, φ(b) = a
1119411⟨a, b | aab=a, abaaa=bb⟩φ(a) = aaaa, φ(b) = a
1120521⟨a, b | ab=aa, aaaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120523⟨a, b | ab=aa, aaaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1120525⟨a, b | ab=aa, aaaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1120527⟨a, b | ab=aa, aaaabb=b⟩φ(a) = a, φ(b) = aaaaaa
1120529⟨a, b | ab=aa, aaabaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120531⟨a, b | ab=aa, aaabab=b⟩φ(a) = a, φ(b) = aaaaaa
1120533⟨a, b | ab=aa, aaabba=b⟩φ(a) = a, φ(b) = aaaaaa
1120535⟨a, b | ab=aa, aaabbb=b⟩φ(a) = a, φ(b) = aaaaaa
1120537⟨a, b | ab=aa, aabaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120539⟨a, b | ab=aa, aabaab=b⟩φ(a) = a, φ(b) = aaaaaa
1120541⟨a, b | ab=aa, aababa=b⟩φ(a) = a, φ(b) = aaaaaa
1120543⟨a, b | ab=aa, aababb=b⟩φ(a) = a, φ(b) = aaaaaa
1120545⟨a, b | ab=aa, aabbaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120547⟨a, b | ab=aa, aabbab=b⟩φ(a) = a, φ(b) = aaaaaa
1120549⟨a, b | ab=aa, aabbba=b⟩φ(a) = a, φ(b) = aaaaaa
1120551⟨a, b | ab=aa, aabbbb=b⟩φ(a) = a, φ(b) = aaaaaa
1120553⟨a, b | ab=aa, abaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120555⟨a, b | ab=aa, abaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1120557⟨a, b | ab=aa, abaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1120559⟨a, b | ab=aa, abaabb=b⟩φ(a) = a, φ(b) = aaaaaa
1120561⟨a, b | ab=aa, ababaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120563⟨a, b | ab=aa, ababab=b⟩φ(a) = a, φ(b) = aaaaaa
1120565⟨a, b | ab=aa, ababba=b⟩φ(a) = a, φ(b) = aaaaaa
1120567⟨a, b | ab=aa, ababbb=b⟩φ(a) = a, φ(b) = aaaaaa
1120569⟨a, b | ab=aa, abbaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120571⟨a, b | ab=aa, abbaab=b⟩φ(a) = a, φ(b) = aaaaaa
1120573⟨a, b | ab=aa, abbaba=b⟩φ(a) = a, φ(b) = aaaaaa
1120575⟨a, b | ab=aa, abbabb=b⟩φ(a) = a, φ(b) = aaaaaa
1120577⟨a, b | ab=aa, abbbaa=b⟩φ(a) = a, φ(b) = aaaaaa
1120579⟨a, b | ab=aa, abbbab=b⟩φ(a) = a, φ(b) = aaaaaa
1120581⟨a, b | ab=aa, abbbba=b⟩φ(a) = a, φ(b) = aaaaaa
1120583⟨a, b | ab=aa, abbbbb=b⟩φ(a) = a, φ(b) = aaaaaa
1124189⟨a, b | aa=b, aaaaaaa=b⟩φ(a) = a, φ(b) = aa
1124727⟨a, b | aa=b, aaaaab=aa⟩φ(a) = a, φ(b) = aa
1124731⟨a, b | aa=b, aaaaba=aa⟩φ(a) = a, φ(b) = aa
1124739⟨a, b | aa=b, aaabaa=aa⟩φ(a) = a, φ(b) = aa